The sum of two numbers is $77$, and their difference is $35$. What are the two numbers?
Answer: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 77}$ ${x-y = 35}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 112 $ $ x = \dfrac{112}{2} $ ${x = 56}$ Now that you know ${x = 56}$ , plug it back into $ {x+y = 77}$ to find $y$ ${(56)}{ + y = 77}$ ${y = 21}$ You can also plug ${x = 56}$ into $ {x-y = 35}$ and get the same answer for $y$ ${(56)}{ - y = 35}$ ${y = 21}$ Therefore, the larger number is $56$, and the smaller number is $21$.